The Animate and the Inanimate in Pure Mathematics: A Modern Reappraisal of William James Sidis's Underrated Viewpoint

William James Sidis’s 1925 work The Animate and the Inanimate introduced a
thermodynamically inspired division of the universe into two regions: one obeying
the conventional second law of thermodynamics (the inanimate) and another where
entropy locally decreases (the animate). Although historically misunderstood and
dismissed as speculative philosophy, his framework can be formalized in modern
mathematical physics and shown to have conceptual parallels with contemporary
theories in non-equilibrium thermodynamics, information theory, and cosmology.
This paper provides a purely mathematical formulation of Sidis’s idea, demon-
strates why it is not inherently incompatible with modern physics, and argues for
its reconsideration, with worked examples and diagrams connecting his concepts to
modern theory.